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On Convergent Dynamic Mode Decomposition and its Equivalence with Occupation Kernel Regression

Moad Abudia, Joel A. Rosenfeld, Rushikesh Kamalapurkar

Abstract

This paper presents a new technique for norm-convergent dynamic mode decomposition of deterministic systems. The developed method utilizes recent results on singular dynamic mode decomposition where it is shown that by appropriate selection of domain and range Hilbert spaces, the Liouville operator (also known as the Koopman generator) can be made to be compact. In this paper, it is shown that by selecting appropriate collections of finite basis functions in the domain and the range, a novel finite-rank representation of the Liouville operator may be obtained. It is also shown that the model resulting from dynamic mode decomposition of the finite-rank representation is closely related to regularized regression using the so-called occupation kernels as basis functions.

On Convergent Dynamic Mode Decomposition and its Equivalence with Occupation Kernel Regression

Abstract

This paper presents a new technique for norm-convergent dynamic mode decomposition of deterministic systems. The developed method utilizes recent results on singular dynamic mode decomposition where it is shown that by appropriate selection of domain and range Hilbert spaces, the Liouville operator (also known as the Koopman generator) can be made to be compact. In this paper, it is shown that by selecting appropriate collections of finite basis functions in the domain and the range, a novel finite-rank representation of the Liouville operator may be obtained. It is also shown that the model resulting from dynamic mode decomposition of the finite-rank representation is closely related to regularized regression using the so-called occupation kernels as basis functions.
Paper Structure (12 sections, 11 theorems, 27 equations, 2 figures)

This paper contains 12 sections, 11 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reproducing Kernel Hilbert Spaces and Dynamic Mode Decomposition
  3. Compact Liouville Operators and Occupation Kernels
  4. Liouville Operators on Real Bergmann-Fock Spaces
  5. Occupation Kernels and Liouville Operators
  6. Finite-rank Representation of the Liouville Operator
  7. Matrix Representation of the Finite-rank Operator
  8. Singular Functions of the Finite-rank Operator
  9. The SLDMD Algorithm
  10. Occupation kernel regression
  11. Numerical Experiments
  12. Conclusion

Key Result

Proposition 1

If $\mu_2 < \mu_1$, then the differential operators $\frac{\partial}{\partial x_i} : F^2_{\mu_1}(\mathbb{R}^n) \to F^2_{\mu_2}(\mathbb{R}^n)$, are compact for $i=1,\ldots,n$.

Figures (2)

  • Figure 1: The blue marks are the mean$\left( \left\vert \tilde{f}(x) \right\vert \right)$ over $x\in[-3,3]$ for different values of $\lambda$ using OKR trained with noisy trajectories. the dashed red line is the mean$\left( \left\vert \tilde{f}(x) \right\vert \right)$ over $x\in[-3,3]$ calculated using the SLDMD method trained with noisy trajectories.
  • Figure 2: The blue marks are the mean$\left( \left\vert \tilde{f}(x) \right\vert \right)$ over $x\in[-3,3]$ for different values of $\lambda$ using OKR trained with noise free trajectories. the dashed red line is the mean$\left( \left\vert \tilde{f}(x) \right\vert \right)$ over $x\in[-3,3]$ calculated using the SLDMD method trained with noise free trajectories.

Theorems & Definitions (11)

  • Proposition 1: SCC.Rosenfeld.Kamalapurkar2023a
  • Proposition 2: SCC.Rosenfeld.Kamalapurkar2023a
  • Theorem 1: SCC.Rosenfeld.Kamalapurkar2023a
  • Proposition 3: SCC.Rosenfeld.Kamalapurkar.ea2022
  • Proposition 4: SCC.Rosenfeld.Kamalapurkar.ea2019
  • Proposition 5: SCC.Rosenfeld.Kamalapurkar.ea2022
  • Proposition 6
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 1 more